Question

# Determine whether a linear , quadratic or exponential function is the best model for the data in each table .

Hint:

### 1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.

2. When the difference between 2 consecutive differences for output values (y values) for a given constant change in the input values (x values) is constant. i.e. dy(n)- dy(n-1) is constant for any value of n, the function is known as a quadratic function.

3. When the ratio between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant i.e. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.

## The correct answer is: The given function is a quadratic function.

### Step-by-step solution:-

From the given table, we observe the following readings-

x1 = 0, y1 = 56;

x2 = 1, y2 = 57;

x3 = 2, y3 = 50;

x4 = 3, y4 = 35;

x5 = 4, y5 = 12

a). Difference between 2 consecutive x values-

dx1 = x2 - x1 = 1 - 0 = 1

dx2 = x3 - x2 = 2 - 1 = 1

dx3 = x4 - x3 = 3 - 2 = 1

dx4 = x5 - x4 = 4 - 3 = 1

Difference between 2 consecutive y values-

dy1 = y2 - y1 = 57 - 56 = 1

dy2 = y3 - y2 = 50 - 57 = -7

dy3 = y4 - y3 = 35 - 50 = -15

dy4 = y5 - y4 = 12 - 35 = -23

We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference are not constant.

Hence, the given function is not a linear function.

b). Now, difference between 2 consecutive differences for y values-

dy2 - dy1 = -7 - 3 = -8

dy3 - dy2 = -15 - (-7) = -15 + 7 = -8

dy4 - dy3 = -23 - (-15) = -23 + 15 = -8

We observe that the difference of differences of 2 consecutive y values are constant i.e. -8.

Hence, the given function is a quadratic function.

Final Answer:-

∴ The given function is a quadratic function.

x2 = 1, y2 = 57;

x3 = 2, y3 = 50;

x4 = 3, y4 = 35;

x5 = 4, y5 = 12

a). Difference between 2 consecutive x values-

dx1 = x2 - x1 = 1 - 0 = 1

dx2 = x3 - x2 = 2 - 1 = 1

dx3 = x4 - x3 = 3 - 2 = 1

dx4 = x5 - x4 = 4 - 3 = 1

Difference between 2 consecutive y values-

dy1 = y2 - y1 = 57 - 56 = 1

dy2 = y3 - y2 = 50 - 57 = -7

dy3 = y4 - y3 = 35 - 50 = -15

dy4 = y5 - y4 = 12 - 35 = -23

We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference are not constant.

Hence, the given function is not a linear function.

b). Now, difference between 2 consecutive differences for y values-

dy2 - dy1 = -7 - 3 = -8

dy3 - dy2 = -15 - (-7) = -15 + 7 = -8

dy4 - dy3 = -23 - (-15) = -23 + 15 = -8

We observe that the difference of differences of 2 consecutive y values are constant i.e. -8.

Hence, the given function is a quadratic function.

Final Answer:-

∴ The given function is a quadratic function.